How can I process neural network with 25000 input nodes?
Yes, it should be no problem.
When you decide to use a CNN, you have to make sure that this makes sense. Another answer mentioned using 3x3 convolutions -- which I would recommend against. For that to work, you would need to turn your vector into a rectangular array, and you would be implying a structure that isn't there.
Use one-dimensional convolutions instead.